Stochastic Mortality Research Articles

Optimal portfolio choice with tontines under systematic longevity risk

AbstractWe derive optimal portfolio choice patterns in retirement (ages 66–105) for a constant relative risk aversion utility maximising investor facing risky capital market returns, stochastic mortality risk, and income-reducing health shocks. Beyond the usual stocks and bonds, the individual can invest his assets in tontines. Tontines are cost-efficient financial contracts providing age-increasing, but volatile cash flows, generated through the pooling of mortality without guarantees, which can help to match increasing financing needs at old ages. We find that a tontine invested in the risk-free asset dominates stock investments for older investors without a bequest motive. However, with a bequest motive, it is optimal to replace the tontine investment over time with traditional financial assets. Our results indicate that early in retirement, a tontine is only an attractive investment option, if the tontine funds are invested in a risky asset. In this case, they crowd out stocks and risk-free bonds in the optimal portfolios of younger investors. Over time, the average optimal portfolio weight of tontines decreases. Introducing systematic mortality risks noticeably reduces the peak allocation to tontines.

Mean reversion in stochastic mortality: why and how?

Life insurance companies use stochastic models to forecast mortality. According to the literature, non-mean reversion models are more suitable for mortality modelling than mean reversion models with a fixed long-term target. In this paper, we adopt stochastic affine processes for the force of mortality and study the impact of adding a time-dependent long-term mean reversion level to two non-mean-reverting processes. We calibrate the models to different generations of the Belgian population and assess these models’ abilities to predict mortality using different statistical methodologies. The backtest shows that the survival curves provided by the mean-reverting processes are closer to reality. Thus, we conclude that incorporating a time-dependent target into these considered models improves their performance significantly.

Estimation of Uncertainty in Mortality Projections Using State-Space Lee-Carter Model

The study develops alternatives of the classical Lee-Carter stochastic mortality model in assessment of uncertainty of mortality rates forecasts. We use the Lee-Carter model expressed as linear Gaussian state-space model or state-space model with Markovian regime-switching to derive coherent estimates of parameters and to introduce additional flexibility required to capture change in trend and non-Gaussian volatility of mortality improvements. For model-fitting, we use a Bayesian Gibbs sampler. We illustrate the application of the models by deriving the confidence intervals of mortality projections using Lithuanian and Swedish data. The results show that state-space model with Markovian regime-switching adequately captures the effect of pandemic, which is present in the Swedish data. However, it is less suitable to model less sharp but more prolonged fluctuations of mortality trends in Lithuania.

Optimal Dynamic Longevity Hedge with Basis Risk

This paper proposes an optimal dynamic strategy for hedging longevity risk in a discrete-time setting. Our proposed hedging strategy relies on standardized mortality-linked securities and minimizes the variance of the hedging error as induced by the population basis risk. While the formulation of our proposed hedging strategy is quite general, we use a stylized pension plan, together with a specified yearly rolling” trading strategy involving q-forwards and a specified stochastic mortality model, to illustrate our proposed strategy. Under these specifications, we show that the resulting hedging problem can be formulated as a stochastic optimal control framework and that a semi-analytic solution can be derived through the Bellman equation. Extensive Monte Carlo studies are conducted to highlight the effectiveness of our proposed hedging strategy. We also consider a scheme to approximate the semi-analytic solution in order to reduce the computational time significantly while still retaining its hedging effectiveness. We benchmark our strategy against the delta” hedging strategy as well as its robustness to q-forwards’ maturity, reference age, and stochastic mortality models. The proposed strategy has many appealing features, including its discrete-time setting which is consistent with market practice and hence conducive to practical implementation, and its generality in that the underlying hedging principle can be applied to other standardized mortality-linked securities and other stochastic models.

Practical Partial Equilibrium Framework for Pricing of Mortality-Linked Instruments in Continuous Time

In this article we demonstrate the practical application of standard partial equilibrium, in backward stochastic differential equation (BSDE) framework, to pricing longevity bonds which are essential for functioning of the life market. The market for mortality linked instruments or so-called life market, is experiencing steady progress in its evolution (see Blake et al. (2018)), however, it is regretfully far from reaching its full potential estimated to be of tens of trillions of dollars (see Michaelson and Mulholland (2014)). Highly illiquid and, compared to equity markets, with a relatively very low number of transactions, the life market is comprised of series of negotiated, high monetary value, over-the-counter transactions between few agents that have different risk preferences. To accommodate these realities we demonstrate an application of a partial equilibrium framework for pricing longevity bonds. We do this under the assumption of stochastic mortality intensity that affects the income of economic agents who trade in risky financial security and longevity bond to maximize their monetary utilities. Thus, by rooting ourselves in a foundational economic principle, as a practical contribution, we find the endogenous equilibrium bond price which is numerically computed. In a realistic setting of two agents in a transaction, numerical experiments confirm the expected intuition of price dependence on model parameters. As a theoretical contribution, we prove that our longevity bond completes the market.

The impact of Hurricane Michael on longleaf pine habitats in Florida

Global biodiversity hotspots (GBHs) are increasingly vulnerable to human stressors such as anthropogenic climate change, which will alter the ecology of these habitats, even where protected. The longleaf pine (Pinus palustris) ecosystem (LPE) of the North American Coastal Plain is a GBH where disturbances are integral for ecosystem maintenance. However, stronger storms due to climate change may be outside their historical norm. In this study, we estimate the extent of Florida LPE that was directly affected by Hurricane Michael in 2018, an unprecedented Category 5 storm. We then leveraged a unique data set in a Before-After study of four sites within this region. We used variable-area transects and generalized linear mixed-effects models to estimate tree densities and logistic regression to estimate mortality by size class. We found at least 28% of the global total remaining extent of LPE was affected in Florida alone. Mortality was highest in medium sized trees (30–45 cm dbh) and ranged from 4.6–15.4% at sites further from the storm center, but increased to 87.8% near the storm center. As the frequency and intensity of extreme events increases, management plans to mitigate climate change need to account for large-scale stochastic mortality events to preserve critical habitats.

Forecasting mortality rates with a general stochastic mortality trend model

This paper presents a model, which can closely estimate the future mortality rates whose efficiency is performed through the comparisons with respect to Lee-Carter and mortality trend models. This general model estimates the logit function of death rate in terms of general tendency of the mortality evolution independent of age, the mortality steepness, additional effects of childhood, youth and old age. Generalized linear model (GLM) is used to estimate the parameters. Moreover, the weighted least square (WLS) and random walk with drift (RWWD) methods are employed to project the future values of the parameters. In order to ensure the stability of the outputs and construct the confidence intervals, Monte Carlo simulation is used. The impact of the proposed model is implemented on USA, France, Italy, Japan and Israel mortality rates for both genders based on their ageing structure. A detailed comparison study is performed to illustrate modified mortality rates on the net single premiums over mortality trend model and Lee-Carter model.

The New Fertility Policy and the Actuarial Balance of China Urban Employee Basic Endowment Insurance Fund Based on Stochastic Mortality Model

This study aims to investigate the impact of China’s new fertility policy on the actuarial balance of its Urban Employee Basic Endowment Insurance (UEBEI) fund, with stochastic mortality model included to address the longevity risk. Combined with the latest UEBEI policy, this paper constructs an actuarial balance model and introduces the growth rate of wage, the age of employment and the age of retirement, the rate of payment, the rate of replacement, the annual rate of pension adjustment, and the population in terms of age into the model, which arise from the rate of payment, average wage, and personal and social factors. This study uses the sixth population census data by age group and gender to employ empirical analysis. The sensitivity analysis of China’s basic pension insurance fund balance is made. It is concluded that the increase in the growth rate of wage caused by social factors, the increase of fund investment returns, the delay of retirement, and the increase in the fund collection rate are all conducive to the sustainability of the UEBEI fund.

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A Quantitative Comparison of Multiple Population Mortality Model on Some East Asian Countries and Regions

This paper reviews the progress of the multiple population mortality model and the defects in parameter estimation and proposes an effective method to improve the performance of the mortality model. We set up a multiple population group, using the data of mainland China, Hong Kong (China), and Japan, to test fitting performance and forecasting performance. Using the TSWLS and TSSVD methods in a multiple population stochastic mortality model has advantages in fitting performance and robustness. In addition, the forecasting value of mortality ratio between any two populations can converge to a fixed constant in a certain time period which obeys the regular of human biological characteristics.

Assessing Longevity Inequality in the U.S.: What Can Be Said About the Future?

This paper investigates mortality inequality across U.S. states by modelling and forecasting mortality rates via a forecast reconciliation approach. Understanding the heterogeneity in state-level mortality experience is of fundamental importance, as it can assist decision making for policymakers, health authorities, as well as local communities who are seeking to reduce inequalities and disparities in life expectancy. A key challenge of multi-population mortality modeling is high dimensionality, and the resulting complex dependence structures across sub-populations. Moreover, when projecting future mortality rates, it is important to ensure that the state-level forecasts are coherent with the national-level forecasts. We address these issues by first obtaining independent state-level forecasts based on classical stochastic mortality models, and then incorporating the dependence structure in the forecast reconciliation process. Both traditional bottom-up reconciliation and the cutting-edge trace minimization reconciliation methods are considered. Based on the U.S. total mortality data for the period 1969{2017, we project the 10-year-ahead mortality rates at both national-level and state-level up to 2027. We find that the geographical inequality in the longevity levels is likely to continue in the future, and the mortality improvement rates will tend to slow down in the coming decades.

Trends and Projections in Breast Cancer Mortality among four Asian countries (1990\u20132017): Evidence from five Stochastic Mortality Models

The current study aimed to explore some important insights into the breast cancer mortality (BCM) trends and projections among four Asian countries by using five advanced stochastic mortality models. BCM data over 28 years from 1990–2017 with ages 20–84 were retrieved from the Global Burden of Disease (GBD) Study 2017 for four Asian countries, namely, China, India, Pakistan, and Thailand. Five stochastic mortality models with the family of generalized age-period-cohort were implemented to find the present and future BCM trends in these four Asian countries. Based on Cairns-Blake-Dowd (CBD) model and Lee-Carter model (LCM), overall, results revealed that BCM increased with the passage of time. Aging factor was the most influential factor of elevated BCM in each Asian country under consideration. Projection of BCM showed that mortality rates might continue to grow with time, especially in older ages in each Asian country under study. The highest forecasted BCM rates were observed in Pakistan as compared to other countries. The obvious increase in BCM suggested that earlier tactics should be implemented to reduce the subsequent morbidity and mortality due to breast cancer. The last but not least, some additional tactics to mitigate the BCM in older ages must be adopted.

Transitory mortality jump modeling with renewal process and its impact on pricing of catastrophic bonds

A number of stochastic mortality models with transitory jump effects have been proposed for the securitization of catastrophic mortality risks. Most of the studies on catastrophic mortality risk modeling assumed that the mortality jumps occur once a year or used a Poisson process for their jump frequencies. Although the timing and the frequency of catastrophic events are unknown, the history of the events might provide information about their future occurrences. In this paper, we propose a specification of the Lee–Carter model by using the renewal process and we assume that the mean time between jump arrivals is no longer constant. Our aim is to find a more realistic mortality model by incorporating the history of catastrophic events. We illustrate the proposed model with mortality data from the US, the UK, Switzerland, France, and Italy. Our proposed model fits the historical data better than the other jump models for all countries. Furthermore, we price hypothetical mortality bonds and show that the renewal process has a significant impact on the estimated prices.

Fitting and forecasting mortality for Assam population by applying the Lee-Carter model

In this paper, we examine the feasibility of using the Lee-Carter model for forecasting the mortality of Assam population. The stochastic mortality model given by Lee and Carter (1992) is used for fitting and forecasting the human mortality of Assam for both the gender. We fit the model to the matrix of Assam death rates based on decade-wise data separately for Assam male and female population in the form of life tables for the period 1995-99 to 2009-13. The parameters of the model are estimated using the Singular Value Decomposition (SVD) method. A time-varying index of mortality is forecasted up to 2030 year using ARIMA model and is used to generate projected life tables.

Stochastic Mortality Modelling for Dependent Coupled Lives

Broken-heart syndrome is the most common form of short-term dependence, inducing a temporary increase in an individual’s force of mortality upon the occurrence of extreme events, such as the loss of a spouse. Socioeconomic influences on bereavement processes allow for suggestion of variability in the significance of short-term dependence between couples in countries of differing levels of economic development. Motivated by analysis of a Ghanaian data set, we propose a stochastic mortality model of the joint mortality of paired lives and the causal relation between their death times, in a less economically developed country than those considered in existing studies. The paired mortality intensities are assumed to be non-mean-reverting Cox–Ingersoll–Ross processes, reflecting the reduced concentration of the initial loss impact apparent in the data set. The effect of the death on the mortality intensity of the surviving spouse is given by a mean-reverting Ornstein–Uhlenbeck process which captures the subsiding nature of the mortality increase characteristic of broken-heart syndrome. Inclusion of a population wide volatility parameter in the Ornstein–Uhlenbeck bereavement process gives rise to a significant non-diversifiable risk, heightening the importance of the dependence assumption in this case. Applying the model proposed to an insurance pricing problem, we obtain the appropriate premium under consideration of dependence between coupled lives through application of the indifference pricing principle.

Introducing and Evaluating a New Multiple-Component Stochastic Mortality Model

This work introduces and evaluates a new multiple-component stochastic mortality model. Our proposal is based on a parameter estimation methodology that aims to reveal significant and distinct age clusters by identifying the optimal number of incorporated period and cohort effects. Our methodology adopts sparse principal component analysis and generalized linear models (GLMs), first introduced in Hatzopoulos and Haberman (2011), and it incorporates several novelties. Precisely, our approach is driven by the unexplained variance ratio (UVR) metric to maximize the captured variance of the mortality data and to regulate the sparsity of the model with the aim of acquiring distinct and significant stochastic components. In this way, our model gains a highly informative structure in an efficient way, and it is able to designate an identified mortality trend to a unique age cluster. We also provide an extensive experimental testbed to evaluate the efficiency of the proposed model in terms of fitting and forecasting performance over several datasets (Greece, England and Wales, France, and Japan), and we compare our results to those of well-known mortality models (Lee-Carter, Renshaw-Haberman, Currie, and Plat). Our model is able to achieve high scores over diverse qualitative and quantitative evaluation metrics and outperforms the rest of the models in the majority of the experiments. Our results show the beneficial characteristics of the proposed model and come into agreement with well-established findings in the mortality literature.

Trends in Canadian Mortality by Pension Level: Evidence from the CPP and QPP

This article looks at the mortality of Canadian pensioners subdivided by pension level using data from the Canada Pension Plan (CPP) and Québec Pension Plan (QPP). Differing pension levels (11 groups) are found to give rise to significant levels of mortality inequality at age 65, with a declining inequality gap as cohorts get older. We also find that levels of inequality have increased slightly over time, and that the QPP pensioners exhibit greater levels of inequality than CPP. Additionally, we find strong, but indirect, evidence among the lowest pension groups for a healthy immigrant effect.We fit a range of multipopulation stochastic mortality models to the CPP and QPP data and find that the common age effect model satisfies a range of quantitative and qualitative criteria. The model allows us to distill further detail from the underlying mortality data as well as provide a coherent basis for forecasting mortality and assessing uncertainty in these forecasts.Lastly, we use clustering methods to consider how significant the differences are between the 11 groups.

Improving HMD Mortality Estimates with HFD Fertility Data

This article aims to improve mortality estimates using fertility data. Estimating the population exposed to risk, such as in the Human Mortality Database (HMD), can suffer from errors for cohorts born in years in which births fluctuate unevenly over the year. When comparing period and cohort mortality tables, we highlight the presence of anomalies in the period tables in the form of isolated cohort effects. Our investigation of the HMD methodology shows that it assumes a uniform distribution of births that is specific to the period tables, which is likely to lead to an asymmetry with the cohort tables. Building on the “Phantoms Never Die” study of Cairns et al. regarding the construction of a “data quality indicator,” we utilize the Human Fertility Database (HFD), which is the perfect counterpart to the HMD in terms of fertility. The indicator is then used to construct corrected period mortality tables for several countries, which are then analyzed from both historical and prospective points of view. The analysis has implications for the reduction of volatility of mortality improvement rates, the use of cohort parameters in stochastic mortality models, and the improved fit of corrected tables by classical mortality models.

The Lee-Carter quantile mortality model

The Lee-Carter (LC) stochastic mortality model has been widely used for making future projections of mortality rates. In the framework of the LC model, the response function is non-linear in parameters. Here, we adapt this LC framework to compute conditional quantiles. The LC quantile model can be defined as quantile non-linear regression conditioned to age and the calendar year. Two strategies for estimating coefficients based on interior-point methods are described. We show that the LC quantile model provides additional information to that furnished by the traditional LC conditional mean. An application to Spanish mortality data is reported.

Combining Investment and Tax Strategies for Optimizing Lifetime Solvency under Uncertain Returns and Mortality

We consider an investor who is looking to maximize their probability of remaining solvent throughout their lifetime by using an algorithm that aims to optimize their investment allocation strategy and optimize their tax strategy for withdrawal allocations between tax deferred accounts (TDAs), Roth accounts, and taxable stock and bond accounts. Our optimization works with stochastic investment returns and stochastic mortality. We find that optimizing the investment strategy (via dynamic programming) has a much larger impact on the investor remaining solvent than optimizing the tax strategy (via Monte Carlo and numerical optimization). This result is key to effectively optimizing both strategies simultaneously. We show that our optimized investment strategy soundly beats a standard target date fund strategy, while our novel optimized tax strategy displays the optimal desired properties suggested by non-stochastic tax optimization research.